Mean Δy profile of long filaments (L_f ≥ 20 Mpc), convolved with the Planck beam at z = 0.4. The dark purple shows the results of gas in the inter-filament medium (i.e. excluding the contribution of galactic haloes), while light purple presents our findings when galactic haloes are included. The lower and upper boundaries (solid purple lines) are computed using Eqs. (8) and (9), respectively. We compare our results from simulations with observations from Tanimura et al. (2020b), which are shown in grey (line and 1σ errors).